Real-time gauge/gravity duality (0805.0150v2)

Abstract: We present a general prescription for the holographic computation of real-time n-point functions in non-trivial states. In QFT such real-time computations involve a choice of a time contour in the complex time plane. The holographic prescription amounts to ``filling in'' this contour with bulk solutions: real segments of the contour are filled in with Lorentzian solutions while imaginary segments are filled in with Riemannian solutions and appropriate matching conditions are imposed at the corners of the contour. We illustrate the general discussion by computing the 2-point function of a scalar operator using this prescription and by showing that this leads to an unambiguous answer with the correct i epsilon insertions.

• The paper introduces a novel piece-wise holographic method to compute real-time n-point functions by matching Lorentzian and Riemannian bulk solutions.
• The paper resolves ambiguities in initial and final data by linking bulk field conditions with the in- and out-states of the boundary QFT.
• The computed scalar two-point function demonstrates the approach’s validity for capturing dynamic phenomena in AdS/CFT duality.

Real-time Gauge/Gravity Duality

The paper "Real-time gauge/gravity duality" by Kostas Skenderis and Balt C. van Rees addresses a pivotal aspect of the AdS/CFT correspondence: the formulation of a holographic prescription for real-time $n$-point functions in quantum field theories (QFT) with holographic duals. The authors present a systematic method to compute these functions in non-trivial states by intricately filling the time contour in the complex plane with appropriate bulk solutions.

The principal challenge tackled in this paper is the necessity to extend the holographic dictionary, traditionally clear from a Euclidean perspective, into the field of Lorentzian spacetimes. In the Euclidean context, the duality framework leverages the AdS/CFT correspondence to explore strong coupling dynamics and gravitational physics by working with equilibrated states and static or time-independent spacetimes. However, for applications in more dynamic scenarios, such as those involving time-dependent probe systems or non-equilibrium states, a robust real-time framework becomes indispensable.

Methodological Details

The authors propose a methodology that treats different segments of the time contour separately: real segments are modeled using Lorentzian solutions, and imaginary components are described with Riemannian solutions, a technique referred to as "piece-wise" holography. These segments are cohesively unified by imposing continuity conditions at the intersection points, ensuring that both the induced fields and their conjugate momenta match across boundaries.

A significant contribution of this work lies in its resolution of ambiguities associated with specifying initial and final data for bulk fields. This concern is pragmatically addressed by linking these initial and final conditions to the in- and out-states of the boundary QFT. More explicitly, the real-time formalism provides a prescription reducing the bulk solution's freedom by adopting specific matching conditions which mirror the expected thermal and non-equilibrium phenomena in the boundary theory.

Quantitative Results

The theoretical formulation is substantiated through the computation of a scalar two-point function as an illustrative example. Calculations conducted strictly adhere to the real-time rules set forth by this new approach, resulting in a well-defined expression replete with the correct $i\epsilon$ prescription, vital for ensuring the anticipated contour integration aligns with time-ordered correlator expectations. This demonstration exemplifies not only the applicable breadth of the prescription but also its capacity to produce consistent physical predictions.

Implications and Future Directions

In terms of implications, this work carries substantive weight for the analysis of time-dependent phenomena in holographic contexts, notably in the examination of systems similar to the quark-gluon plasma—a domain where real-time dynamics are crucial. The methodology also adds depth to the understanding of how AdS holography could encode information about events beyond event horizons in bulk spacetime geometries, suggesting avenues for future exploration of quantum gravitational phenomena.

The authors point out that while they have developed the formalism for a particular class of contour specifically addressing vacuum-to-vacuum transition amplitudes, the discussion can be broadened to encompass more complex scenarios including thermal ensembles and closed in-in contours. Anticipated extensions, encompassing various geometric settings or more complex bulk field dynamics, are probable future developments. The examination of the coupling between asymptotically AdS spaces and the holographic boundary dynamics within this framework could also inform investigations into holographic renormalization group flows, horizon complementarity, and quantum entanglement entropy dynamics.

In conclusion, the paper establishes a foundational real-time holographic prescription for AdS/CFT dualities that enriches the toolkit available for contemporary quantum field theorists and string theorists seeking deeper insights into non-static, dynamical quantum states. Through ensuring correspondence between dynamical states and partition function evaluations across compatible geodesic networks, the approach reinforces confidence in applying gauge/string dualities to real-world phenomenological problems where time evolution plays a critical role.